Correlations in strong-field-emitted ultrashort electron pulses from metal needle tips

When two electrons are emitted from a metal needle tip with the help of femtosecond laser pulses, they show a strong anticorrelation signal in the energy domain. Depending on the wavelength and intensity of the driving laser pulses, the electron emission process can be either in a perturbative regime, like single- or multi-photon photoemission, or in the strong-field regime, where emission is dominated by the instantaneous electric field of the laser pulse, or in the intermediate regime. Here, we report on the two-electron anticorrelation signal and how it evolves from the multiphoton toward the strong-field emission regime. We show that in both cases, the resulting anticorrelation signal can be well explained by semi-classical simulations using a point-particle model, thus the dynamics is dominated by the center-of-mass dynamics of the individual electrons. However, the actual emission process of multiple interacting electrons requires improved quantum mechanical models and therefore remains the subject of future work. This paper is part of the Special Topic Collection: papers from the 31th Annual International Laser Physics Workshop 2023 (LPHYS 2023).


Introduction
Ultrashort laser pulses interacting with atoms in the gas phase has evolved to modern attosecond physics, culminating in the 2023 Nobel Prize in physics.Already during the first experiments on ultrashort laser pulses interacting with gas atoms, it was recognized that the number of detected doubly charged ions was orders of magnitude higher than expected at the time Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.[1].This observation could only be fully understood by correlation experiments of both emitted electrons [2], finally leading to the full understanding of the process we know today as non-sequential double ionization (NSDI) [3].For such experiments, the intensity of the laser pulses is typically in the order of 1 × 10 14 W cm −2 or above, leading to electron emission in the so-called strong-field regime.This regime is of high relevance, as it opens the door to the attosecond time scale [4,5].
In the last two decades, strong-field electron emission has also been achieved from solid-state targets, for example at metal needle tips [6][7][8][9][10][11][12] or nano-spheres [13].Like for atoms, emitted electrons can undergo an elastic rescattering process at their origin as they are driven in the strong-laser field.Recently, also in such system electron-electron correlations were investigated, specifically correlations in electron beams emitted from metal needle tips [14,15].In these experiments, the probability distribution of two electrons emitted by one pulse shows a clear dip around an energy difference of ∆E = 0.This indicates clearly an energy anticorrelation, which stems from dynamic Coulomb-interactions of the electrons subsequent to their emission.Because the emission is strongly confined in space and time to the nanometer and femtosecond scale two electrons suffice to observe this effect.These interactions thus provide a possibility to manipulate the electron number statistics of the beam by filtering, enabling sub-Poissonian electron beams.
The electron emission process depends on the laser intensity for laser pulses at 800 nm.It can either be a perturbative multi-photon process for low intensities that transits to the strong-field emission regime with increasing intensity [16].Compared to atoms, the probability of emitting two electrons from a single needle tip is higher at low intensities, which is why we can investigate correlations in both the multiphoton as well as the strong-field regime at tips.For experiments using laser pulses at 800 nm it was shown that the depth of the anticorrelation gap decreases as a function of intensity [14].
In the following, we will revisit the intensity dependence of the anticorrelation signal and revisit the interplay between electron-electron interaction and the optical near-field from low fields toward the strong-field regime, as shown in [14].We show here that the observed intensity dependence of the gap is well reproduced by a semi-classical simulation, in both regimes, the perturbative as well as the strong-field regime.

Experiment
The experimental setup is shown in figure 1(a).A tungsten needle tip with a radius of 10-15 nm, in this case biased by U tip = −30 V, is placed inside an ultra-high vacuum chamber (not shown).We focus ultrashort laser pulses with a duration of ∼12 fs, a repetition rate of 200 kHz and a central wavelength of 800 nm on the tip to trigger electron emission.We increase the optical near-field intensity at the tip apex from 0.8 × 10 13 W cm −2 to 2.3 × 10 13 W cm −2 by changing the incident pulse energy with a neutral density wheel.This intensity already includes a field enhancement factor of 3.7, determined from the cut-off of the plateau of the electron spectra [17,18].Within this range of intensities we capture the transition region between the multiphoton regime toward the strong-field regime.In the strong-field regime, the electrons can either be emitted directly, or undergo a rescattering process with the metal tip, depending on the phase of the laser field when they are emitted.In this process, the electrons can gain additional energy, leading to the aforementioned plateau in their energy spectrum [8,9,19,20].Thus there are two mechanisms, the strongly driven electron in the optical near field and Coulomb interaction, which can strongly influence the final energy of two-electron events, shown schematically in figure 1(b).
The electrons are detected by a delay-line detector, capable of measuring multiple electrons [21,22].At this detector, the electrons hit a multi-channel plate, get multiplied and travel to a set of wire-grids.There they induce voltage pulses that are measured at each end of each wire.In such a way, the position and time-of-flight can simultaneously be measured for two electrons emitted by one laser pulse.Before the detection, the emitted electron beam gets magnified by a pair of quadrupole lenses, both to enlarge the central part of the beam and to reduce the effects of the dead-radius of the detector.Details of the experimental setup are shown in [14].
After the measurement, we select all events where two electrons were recorded and calculate the energy of these electrons from their position and time-of-flight information.In figure 2(a) we show the energy difference spectra of the two-electron events as a function of near-field intensity from low intensities (blue) to high intensities (red).We can see that the dip becomes less pronounced for larger intensities.Furthermore, a shoulder arises at the sides of the spectra, which stems from the plateau of the rescattered electrons.Figure 2(a) shows the same data as published in [14].

Simulation
We simulate the intensity-dependent spectra using a semiclassical simulation, based on the simple man's model, also called three-step model [23,24].In this simulation, electrons are propagated as point-particles, iterated for different initial conditions.For the initial spatial position, we chose a 2D Gaussian distribution mapped on a hemisphere representing the apex of the tip.The initial temporal distribution is given by the instantaneous rate model of electron emission in a laser field, formulated in the paper of Ivanov et al [25].Details of the formula we used are given in the method section of [14].We chose parameters close to the experiment, namely a tip radius of 15 nm, a pulse duration of 12.5 fs and an intensity range from 0.4 × 10 13 W cm −2 to 4.1 × 10 13 W cm −2 at a central wavelength of 800 nm.In addition, we mark the electrons that have rescattered to distinguish them from directly emitted electrons, as shown later.To account for the finite detector resolution in the experiment as well as the smearedout nature of a quantum mechanical wave packet, we convoluted the resulting energy difference spectra with a Gaussian function with a normal deviation of 0.7 eV.Further, we needed to take into account a bigger detector radius of 60 mm instead of 40 mm to match the simulations with the experiment.The resulting energy difference spectra are shown in figure 2(b).Just as for the experimental data, we see that the gap shrinks for increasing intensity and a shoulder develops in the spectra.We can therefore show that for a large range of intensities, from the multiphoton to the strong-field regime, the experimental data are well described by our simulation.

Discussion
We compare the experimental and simulated data by means of the depth of the gap that we quantify by the visibility V of the energy gap, given by V = Imax−I min Imax+I min .We call V the repulsion visibility.Here I max is the (mean) maximum of the two main peaks next to the gap and I min is the count value in the gap between the peaks.Experimental and theory data are shown in figure 2(c), where the blue data corresponds to the simulation and the orange data is extracted from the experiment.Both curves show a strong decrease of repulsion visibility as a function of intensity.The simulation further shows a very good agreement with the experimental data using the reasonable smoothing function of 0.7 eV as mentioned above.The repulsion visibility is in particular interesting since it represents the suppression of the two-electron events with the same energy.It is thus a more or less direct measure of the sub-Poissonian statistics.Further, a pronounced gap allows disentangling two-electron events from single electron events more efficiently, which is important for heralded imaging schemes with electrons.Driven by these two possible applications, it is therefore important to know the influences of various parameters on the energy gap.The intensity is a special parameter here, as the additional driving of the electrons by the optical near-field changes the emission process itself and thus the initial conditions for the further propagation.
In the present work we show that in this transition of emission regimes, a 3D semi-classical simulation allows us to extract the most important features of the experiment, thus such a model can be used for engineering the anticorrelation gap in simulation.Although full 3D quantum simulations might allow deeper insights for such problems, semi-classical simulations are often used in numerical simulations of similar problems, such as NSDI, where they have yielded even quantitative insights [26,27].Further, quantum simulations like the time-dependent Schrödinger equation or the time-dependent density functional theory are usually much more computationally intensive.
As we can see in figure 2(c), both simulation and experiment show a repulsion visibility of V ≈ 0.5 for an intensity of 0.8 × 10 13 W cm −2 that decreases to a value of V ≈ 0.3 at 2.3 × 10 13 W cm −2 .These intensities correspond to ponderomotive energies of 0.5 eV and 1.3 eV, thus the spectrum at the lowest intensity does not show any plateau yet, while the plateau is clearly pronounced for the value of the highest intensity, indicating the strong driving of the electrons in the near-field of the tip.Note that the mean energy splitting, given by the distance between the two peaks in the energy difference spectra, stays almost constant as function of intensity for the experimental data as well as the simulation.
Our data shows that in order to obtain a high two electron event rate while maintaining a high repulsion visibility, staying in the perturbative regime is preferable.For this, using a smaller wavelength, i.e. reducing the non-linearity of the emission, is advantageous.For smaller wavelengths ponderomotive effects are reduced for the same intensity.Alternatively when changing the wavelength is not possible, tuning the size of the emitter in combination with spatial filtering could be another route.At a larger tip, more electrons can be emitted already at lower intensities, but a filter aperture should become necessary, as the repulsion energy will also be reduced, thus only closely emitted electrons show a strong anticorrelation.
In figure 2(d) we show the emission times of the electron that is emitted first (blue) and the corresponding second electron (orange) in a two-electron event when the electrons are filtered spatially.For that purpose we limit the virtual detector of the electrons to a radius of 60 mm, omitting all electrons that are outside of this area.This spatial filtering is equivalent to placing an aperture in the beam path.Furthermore, we indicate in light green the times when direct electrons are born, and in light blue the emission time of rescattered electrons.We observe that the first electron is ∼51% more likely to be a rescattered one than the second electron, due to Coulomb repulsion effects.Without Coulomb repulsion, both distributions would be equal.This shows that especially those two-electron events pass spatially through an aperture in which the first electron has gained more energy and thus longitudinal effects outweigh transverse repulsion effects.Note that in this case an aperture does not actively change the trajectories.Thus the combination of Coulomb interactions and strong-field electron emission could act as a tuning knob to separate direct and rescattered electrons by their emission time, which might be advantageous in the future.Additionally, increasing the intensity reduces the effective emission time window, which again could, in combination with Coulombbased anticorrelations, lead to another possibility to engineer electron beam statistics.This could be particularly interesting for larger central wavelengths, as then the emission times are shortened again and ponderomotive effects are further enhanced.

Summary
We have presented semi-classical point particle simulations of two-electron emission events from nano-scale sources for different laser intensities, ranging from the perturbative to the strong-field electron emission regime.The evaluation of the repulsion visibility showed an excellent agreement between simulation and experiment, indicating that a semi-classical description is sufficient for this case.Although it would be highly beneficial to obtain a full 3D quantum two-particle simulation in the future, the present possibility of semi-classical simulations offers a suitable tool to engineer the relevant parameters for electron number statistics tuning.We envision and have shown first steps on using strong-field emission physics in combination with Coulomb repulsion to enable future complex manipulation of pulsed electron beams.

Figure 1 .
Figure 1.(a) Experimental setup.An ultrafast laser pulse is focused on a metal needle tip.Electrons are emitted and accelerated toward a multi-hit capable delay-line detector.(b) The incident laser intensity is varied leading to different electron trajectories, that change the mutual interaction between both electrons (yellow arrow).

Figure 2 .
Figure 2. Comparison of experiment and simulation.(a) Experimental data of energy-difference spectra of the two-electron events as a function of laser intensity, as first shown in [14].(b) Simulated energy-difference spectra using our semi-classical point-particle simulation using the same parameter as in the experiment for the tip and the laser pulses.(c) Repulsion visibility of the simulated (blue) and the experimental (orange) data.The shading in the background indicates the transition from the perturbative (white) to the strong-field regime (red).(d) Starting times of the first (blue) and second (orange) electron for simulated two-electron events within a detector radius of 60 mm.Electrons emitted in green shaded areas are direct electrons whereas electrons emitted in the blue area are rescattered ones.